Generation of analytic semi-groups by second-order differential operators with nonseparated boundary conditions

Abstract

We investigate under which boundary conditions a second order differential operator of the form Lu = u″ + q1(x)u′ + q0(x)u generates an analytic semi-group in Lp(a, b), 1 ≤ p ≤ ∞. The boundary conditions are not supposed to be separated: Bi(u) ≡ aiu(a) + biu′(a) + ciu(b) + diu′(b) = 0, i = 1, 2, and this allows us to obtain quite general results. The generation of analytic semi-groups is proved by showing the estimate ∥R(λ : L)∥ ≤ M∣λ∣-1 for the resolvent operator in a suitable sector of the complex plane. © 2000 Rocky Mountain Mathematics Consortium.